Graphs With Least Eigenvalue —2: A New Proof of the 31 Forbidden Subgraphs Theorem

Abstract

Generalized line graphs were introduced by Hoffman Proc. Calgary Internat. Conf. on Combinatorial Structures and their applications, Gordon and Breach, New York (1970); they were characterized in 1980 by a collection of 31 forbidden induced subgraphs, obtained independently by Cvetković et al., Comptes Rendus Math. Rep. Acad. Sci. Canada (1980) and S. B. Rao et al., Proc. Second Symp., Indian Statistical Institute, Calcutta, Lecture Notes in Math., (1981). Here a short new proof of this characterization theorem is given, based on an edge-colouring technique.

Some observations on the smallest adjacency eigenvalue of a graph

“…When µ = 2, we again have λ(L( Ĝ)) ≥ −2. The graphs L( Ĝ) with µ = 2 are the generalized line graphs of Hoffman [25] (see also [18,21] or [20, p.6]). The usual proofs that λ(L( Ĝ)) ≥ −2 for a generalized line graph L( Ĝ) employ modifications of the vertex-edge incidence matrix of Ĝ [25], [20, p.6].…”

Characterizations of the family of all generalized line graphs-finite and infinite- and classification of the family of all graphs whose least eigenvalues \ge - 2

“…If u is a vertex of two members of F, say F α , F β , then let ξ(u) = α + β. Suppose that u belongs to exactly one member, say [4,5] and five different methods of computing M f have been found in [4,9,13,5,14]. Countably infinite graphs in L also have been studied: in [10], it has been shown that any countably infinite connected graph with least eigenvalue −2 is a generalized line graph and in [11], all countably infinite connected graphs with least eigenvalues > −2 have been determined.…”

“…In [14], M f has been computed by using the finite version of Theorem 10 but not directly. (See [4,9,5] for different methods.) This has been done in [13] and [3] by using results for finite graphs which are similar to Theorem 10.…”

Equivalence of four descriptions of generalized line graphs

“…Let M be the class of all minimal nongeneralized line graphs; i.e., a graph G belongs to M if and only if G / ∈ G but every proper induced subgraph of G belongs to G. Since M determines G-a graph G belongs to G if and only if no induced subgraph of G belongs to M-computation of M has received considerable attention. By using the spectral properties and the representations of the graphs in L 2 and Theorem 4, M has been obtained in [3] whereas this has been accomplished in [7] by exploring the structural properties of the graphs in G. The study of G and M has been revived in [4] where one more approach has been described to construct M. Earlier in [9], M has been found by using a notion called "symmetric set in a graph" which involves a graph involution. From this, the key notion of this article has emerged; but the latter is simpler: it is based only on some basic properties of generalized line graphs.…”